As long as algebra and geometry travelled separate paths their advance was slow and their applications limited. But when these two sciences joined company, they drew from each other fresh vitality and thenceforward marched on at a rapid pace towards perfection. It is to Descartes that we owe the application of algebra to geometry,-an application which has furnished the key to the greatest discoveries in all branches of mathematics.
René DescartesRelated topics
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Cardano was a man of universal interests, and much of his ability must have been inherited from his father, Fazio Cardano... a lawyer... but also deeply steeped in the medical sciences, mathematics, and all kinds of occult lore... he had... a high reputation as a scholar in his native town; even Leonardo da Vinci notes several times that he consulted Messer Fazio on geometric questions... he was appointed as a public lecturer in geometry.
Gerolamo Cardano
There is only one other survey, Datta and Singh's 1938 History of Hindu Mathematics, recently reprinted but very hard to obtain in the West (I found a copy in a small specialized bookstore in Chennai). They describe in some detail the Indian work in arithmetic and algebra and, supplemented by the equally hard to find Geometry in Ancient and Medieval India by Sarasvati Amma (1979), one can get an overview of most topics.
David Mumford
In my own time there have been inventions of this sort, transparent windows tubes for diffusing warmth equally through all parts of a building short-hand, which has been carried to such a perfection that a writer can keep pace with the most rapid speaker. But the inventing of such things is drudgery for the lowest slaves philosophy lies deeper. It is not her office to teach men how to use their hands. The object of her lessons is to form the soul.
Seneca
Looking at them, it does not seem as though even progress in intellect, slow as that also is, made the pace very much more rapid. When we look at those whose intellect is scarcely developed, they seem after each day of life to sink to sleep almost on the place they occupied the day before; and when we glance over those who are more highly evolved so far as intellect is concerned, they too are travelling very very slowly, and seem to make small progress in each day of life.
Annie Besant
It is less than four years since cohomological methods (i. e. methods of Homological Algebra) were introduced into Algebraic Geometry in Serre's fundamental paper[11], and it seems certain that they are to overflow the part of mathematics in the coming years, from the foundations up to the most advanced parts. ... [11] Serre, J. P. Faisceaux algébriques cohérents. Ann. Math. (2), 6, 197–278.
Alexander Grothendieck